Greetings to the Universe.

A golden phonograph record was attached to each of the Voyager spacecraft that were launched almost 25 years ago. One of the purposes was to send a message to extraterrestrials who might find the spacecraft as the spacecraft journeyed through interstellar space. In addition to pictures and music and sounds from earth, greetings in 55 languages were included.

NASA asked Dr Carl Sagan of Cornell University to assemble a greeting and gave him the freedom to choose the format and what would be included. Because of the launch schedule, Sagan (and those he got to help him) was not given a lot of time. Linda Salzman Sagan was given the task of assembling the greetings.

The story behind the creation of the “interstellar message” is chronicled in the book, “Murmurs of Earth”, by Carl Sagan, et al. Unfortunately, not much information is given about the individual speakers. Many of the speakers were from Cornell University and the surrounding communities. They were given no instructions on what to say other than that it was to be a greeting to possible extraterrestrials and that it must be brief.

There are excerpts from the book, and, most importantly, a selection of audio clips of the greetings, from Akkadian (“May all be very well”) to Wu (“Best wishes to you all”). I hope the study of Earth languages will distract our new insect overlords from their plans to enslave us.

The Turkish construction, ironically, contains no reference to heads of any number. The phrase, “sabah şerifleriniz hayrolsun”, is more along the lines of “bless your wonderful morning”. According to Ekşi Sözlük, it’s a modernization of the Ottoman greeting “sabah-ı şerifleriniz hayır olsun”, and seems to be most well known as “that Turkish phrase they sent into space”.

Also of note, the Semitic root s-l-m shows up in the greetings in: Aramaic, Hebrew, Indonesian, Urdu, and Sumerian which borrowed it from Akkadian as the stative verb “silim”, meaning “to be healthy”.

I hope the study of Earth languages will distract our new insect overlords from their plans to enslave us.

Unfortunately it won’t distract them for very long; I have it on good authority that a Martian would conclude that all Terran languages are mere dialects—worse yet, the underlying grammar is universal, so unless our invaders come from another dimension they probably already speak Hittite, more or less.

The Japanese greeting is almost comically casual, even in the original, like we’ve just run into the aliens in the supermarket or something. The aliens are going to turn up and be like “Hey, where are the Japanese? They seem to have the least attachment to politeness and formality in social interactions, we’d like to hang out.”

Suppose I were a linguist and found a multilingual inscription in 55 unknown languages, but with each version limited to one or two sentences of somewhat different meaning, thus forever precluding learning anything about any of these languages. I would bear a very deep grudge against the person who had assembled that collection.

Suppose I were a linguist and found a multilingual inscription in 55 unknown languages, but with each version limited to one or two sentences of somewhat different meaning, thus forever precluding learning anything about any of these languages. I would bear a very deep grudge against the person who had assembled that collection.

The image that immediately comes to mind is the arrival of an enormous alien battle fleet, posed in orbit above Earth and threatening to blast us into vapour unless we explain immediately how Russian noun endings work. “IT’S REALLY BEEN BUGGING US.” Somewhat like Star Trek IV: The Voyage Home but with academic linguists instead of humpback whales.

are a thing in Graph Theory. Working them out involves extraordinary complex and intensive work. Very few have been calculated.

Paul Erdős, the brilliant but eccentric (even given their cultural norms) mathematician, remarked that if an immensely powerful alien being threatened to destroy the earth unless we told it the exact value of R(5,5) we should set all the computers on earth to work on the answer; if it asked for R(6,6) we should try to destroy the alien.

The question of Russian morphology may be analogous.

(Ramsey was a close friend of Wittgenstein and brother of the Archbishop of Canterbury. This is doubtless significant.)

What I find remarkable is the size differential between the upper and lower bounds on some quantities in Ramsey theory. The upper bounds can be so stupefyingly large that special notations are needed to express them, while the lower bound might be, say, 6.

If you’re referring to the Graham number problem, the upper bound had since been revised to something that only needs a very small amount of special notation (mind you, it’s still stupefyingly large), and the lower bound is now 13.

Graham himself believed 6 to be the correct answer; IIRC, the modern estimate is that the true answer is probably closer to 100, but still a fairly small number.

@January First-of-May: I did know that actually, but 6 is funnier. Moreover, the “famously” immense upper bound was really a simpler upper bound for the real (more complicated) upper bound. They’re all still really big numbers though.

I seem to remember reading somewhere that Bourbaki’s Éléments takes a couple of hundred pages to deal with “one.” Still, once you’ve got that far, Graham’s number is trivial. Even “six” becomes tractable in principle, though it’s not easy to envisage a practical application of the concept.

note the footnote — several dozens of thousands of symbols turn out to be an underestimation, according to Wikipedia s.v. Epsilon notation it’s more like four and a half trillion once you expand out all the quantifiers.

But all it really says is that 1 denotes a single set, the outcome of Hilbert’s epsilon operator when considering the predicate on the universe of sets that a bijection can be found between the set in question and the fixed set that has the empty set as its only member. That, boys and girls, is the kind of mess you get yourself in when trying to avoid the Axiom of Choice.

Given any set of foundations of theories, providing they are all founded on at least one axiom, it’s always possible to found a new theory based on exactly one axiom from the foundation of each of the other theories The axiom of choice of axioms states that this holds even without any criteria for the picking, as long as one can make a choice.

Banach-Tarsky tells us that in any wowen fabric, you can always pull out the threads and rearrange them into two identical copies of the original fabric. This is usually the point when I’m allowed to leave the curtain shop.

There’s a Feynman story, possibly apocryphal, about Banach-Tarski. He went to a lecture at Caltech about B-T, which had a jokey subtitle along the lines of “how to wrap the sun in the peel of one orange.” So the lecturer explained how you slice and dice and rearrange to make one sphere into a bigger sphere, and finally Feynman interrupts and says, when you are going to explain how to wrap the sun in an orange peel? The lecturer is a little flustered and starts going back to his explanations, and then Mr Smart Alec pipes up and says, Oh, I thought you meant an actual orange.

Update: On of the SYJF anecdotes has Feynman explaining that he can only follow mathematical arguments by keeping a physical model in his head. So his question was probably fairly legitimate for Feynman rather than smartass.

No, the irrationals are measurable. They are the complement of a countable union of one point sets, which automatically has Lebesgue measure zero. It takes the Axiom of Choice to get the set I’m talking about, which had outer measure 1 and a complement with outer measure 1 also.

I posted a query at Math With Bad Drawings (another part of the Elven realms) and got this reply: “[My wife]’s a harmonic analyst – I’m sure “monstrously big” overstates it, but when you traffic in epsilons, even the small positive integers feel pretty enormous.” I suppose these are the same epsilons being learnedly discussed above? The only epsilon I know about is “your number however small”, as opposed to delta, which is “my number that is even smaller.” Or are these somehow the same as well?

Harmonic analysis is typically “micro-local,” which sounds like it should involve numbers even smaller than the epsilons coming up in limits. However, that’s not what it means. Despite having taken partial differential equations, I never did figure out what the “micro-” part was supposed to be. However, Wikipedia says:

The term microlocal implies localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point.

@JC, harmonic analysis would be part of, well, analysis, and use the sort of deltas and epsilons that we know from high school. Hilbert’s epsilon is an operator used in investigations of the foundations of mathematics, and has nothing in particular to do with numbers — it just happens to be used by Bourbaki in their definition of 1. (They spell it with a tau, however).

I can try to illustrate how Hilbert’s epsilon is used. If you have a predicate with an unbound variable and use epsilon to bind it, the resulting term denotes an object that either makes the predicate true, is one exists, or else some object that (necessarily) doesn’t. Importantly, the object is not random, the same formal expression of a predicate will always result in the same object.

So say that you want to express the proposition that all swans are white. Your predicate will be ‘x is a swan and x is not white’ (the negation of ‘x is a swan implies that x is white’); if Hilbert’s epsilon gives you back a non-white swan, you know your proposition is false; if you get something else, a white swan or an old shoe for instance, you know that no non-white swans exist and your proposition is true.

Yeah, I may be completely wrong, but it struck me as a version of Zeno’s paradox: Making an argument based on approaching but never reaching infinity, and ignoring that a complimentary entity approaches zero at the same time. Zeno ignored time, Banach-Tarski density. But that’s probably me being too bound to the physical world. In this ideal mathematical space, density is being ignored because everything is infinitely dense (or not dense at all, which is effectively the same thing), just as in a world without time Zeno’s paradox would be entirely valid, if meaningless (but that’s me being bound to the physical world again).

A story told on Usenet concerning Banach: There was a mathematics student who selected Russian for his language requirement. But when he showed up to the exam, it turned out they didn’t have one for Russian. So they found someone Russian, who handed him a paper to translate. He stared at it for an hour and the only thing he could read was the title: “The Hahn-Banach Theorem”. Well, hell, he said, I know how to prove that, so he wrote out a proof. The examiner said, “Your translation was pretty free, but you clearly got the gist of it,” so he passed.

Actually, these kinds of fractal structures are not as devoid of real physics content as you might think. According to Liouville’s Theorem (second of that name), the phase space occupied by the possible states of a non-dissipative mechanical system is constant in time. This is a problem for statistical mechanics, since Boltzmann posited that the logarithm of the phase space volume was the entropy (something which, up to that point, had been considered purely a mathematical abstraction). Boltzmann showed that, under reasonable conditions, the entropy was never decreasing, but Liouville’s Theorem seemed to show it could never be increasing either.

So what happens to the phase space when, say, a gas is allowed to expand freely into a larger vessel? Within a very short time (10 collision times is typically enough), Liouville’s phase space has been distorted into a fractal shape. Technically, it’s volume has not increased, but it has snaked around through all of the larger available phase space. So, for macroscopic purposes, it appears that the occupied phase space has increased, as the Second Law of Thermodynamics says it should.

The fractals in these cases do not really have structure down to arbitrarily small scales, like the ones in the Banach-Tarski paradox. However, the pure mathematics of idealized fractals did make an important contribution to the understanding of this thorny physical problem.

There is also another, historical parallel between set theory in math and statistical mechanics in physics. The two fields were essentially created by Cantor and Boltzmann, respectively, working all alone. While neither area was without intellectual antecedents, they required hugely original intellectual leaps to become new and fundamental realms of research.* They were working at exactly the same (just a year apart in age), and the difficulty they had getting their ideas accepted early on led each of them into major bouts of depression. And after their deaths, their foundational work could be memorialized at or near their grave sites with a single concise equation each.

*The best definition of a genius that I have ever heard is that it is someone who answers questions that nobody before had even thought to ask.

Well, it is in the nature of mathematical exposition to use notation that has been shown to be reducible to a more primitive formalism. In fact Bourbaki’s definition of one is τ_Z(Eq( { ∅ }, Z)) which is pretty tractable for humans because we don’t need to work through the underlying definitions to work with it; the expanded form is just a curiosity and huge number of symbols is very much a consequence of expanding the above with the general formula for ‘Eq’ (cardinal equivalence) — if you actually needed to work with the primitive form it could easily be reduced to something much much smaller.

2, now…

However, Bourbaki’s choice of formalism is not popular any more. Otherwise people would have fun trying to prove bounds on the minimal length of a primitive term equal to 1, maybe even managing to fit one on a sheet of paper, instead of using big numbers to make it look silly. (EDIT: Actually there’s a formula for a 176-symbol primitive notation for 1 in section 8 of Piotr’s link. And the next section has an anti-Bourbachiste rant eerily reminiscent of anti-Chomskyan dittos seen in linguistics).

I don’t think the Timecube guy has actually tried a proof in the first place…

Anyway, time doesn’t come in arbitrarily small amounts any more than matter does: quantum uncertainty is such that it doesn’t make sense to deal with amounts smaller than a Planck time (10^-43 seconds).

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